--- trunk/chrisDissertation/Ice.tex	2006/09/08 16:13:21	3003
+++ trunk/chrisDissertation/Ice.tex	2006/09/18 20:32:11	3004
@@ -220,6 +220,14 @@ methods.\cite{Baez95b}
 parameter.  This method has been shown to be reversible and provide
 results in excellent agreement with other established
 methods.\cite{Baez95b}
+
+The Helmholtz free energy error was determined in the same manner in
+both the solid and the liquid free energy calculations . At each point
+along the integration path, we calculated the standard deviation of
+the potential energy difference. Addition or subtraction of these
+values to each of their respective points and integrating the curve
+again provides the upper and lower bounds of the uncertainty in the
+Helmholtz free energy.
 
 Near the cutoff radius ($0.85 * r_{cut}$), charge, dipole, and
 Lennard-Jones interactions were gradually reduced by a cubic switching
@@ -427,9 +435,9 @@ TIP5P-E & & & & - & \\
 \cmidrule(lr){2-6}
 & \multicolumn{5}{c}{(kcal mol$^{-1}$)} \\
 \midrule 
-TIP5P-E & & & & - & \\
-TIP4P-Ew & & -13.09(3) & & - & -12.98(3) \\
-SPC/E & -12.99(3) & -13.00(3) & & - & -12.99(3) \\
+TIP5P-E & -11.98(4) & -11.96(4) & & - & -11.95(3) \\
+TIP4P-Ew & -13.11(3) & -13.09(3) & -12.97(3) & - & -12.98(3) \\
+SPC/E & -12.99(3) & -13.00(3) & -13.03(3) & - & -12.99(3) \\
 SSD/RF & -11.83(3) & -11.66(4) & -12.32(3) & -12.39(3) & - \\
 TRED & -12.61(3) & -12.43(3) & -12.89(3) & -13.12(3) & - \\
 \end{tabular}
@@ -442,9 +450,36 @@ ice B is destabilized such that it is no longer nearly
 the same fashion; however Ice-$i$ and ice B are quite a bit closer in
 free energy (nearly isoenergetic). The free energy differences between
 ice polymorphs for TRED water parallel SSD/RF, with the exception that
-ice B is destabilized such that it is no longer nearly isoenergetic
-with Ice-$i$.
+ice B is destabilized such that it is not very close to Ice-$i$. The
+SPC/E results really show the near isoenergetic behavior when using
+the electrostatics correction. Ice B has the lowest Helmholtz free
+energy; however, all the polymorph results overlap within error.
 
+The most interesting results from these calculations come from the
+more expensive TIP4P-Ew and TIP5P-E results. Both of these models were
+optimized for use with an electrostatic correction and are
+geometrically arranged to mimic water following two different
+ideas. In TIP5P-E, the primary location for the negative charge in the
+molecule is assigned to the lone-pairs of the oxygen, while TIP4P-Ew
+places the negative charge near the center-of-mass along the H-O-H
+bisector. There is some debate as to which is the proper choice for
+the negative charge location, and this has in part led to a six-site
+water model that balances both of these options.\cite{Vega05,Nada03}
+The limited results in table \ref{tab:dampedFreeEnergy} support the
+results of Vega {\it et al.}, which indicate the TIP4P charge location
+geometry is more physically valid.\cite{Vega05} With the TIP4P-Ew
+water model, the experimentally observed polymorph (ice
+I$_\textrm{h}$) is the preferred form with ice I$_\textrm{c}$ slightly
+higher in energy, though overlapping within error, and the less
+realistic ice B and Ice-$i^\prime$ are destabilized relative to these
+polymorphs. TIP5P-E shows similar behavior to SPC/E, where there is no
+real free energy distinction between the various polymorphs and lend
+credence to other results indicating the preferred form of TIP5P at
+1~atm is a structure similar to ice B.\cite{Yamada02,Vega05,Abascal05}
+These results indicate that TIP4P-Ew is a better mimic of real water
+than these other models when studying crystallization and solid forms
+of water.
+
 \section{Conclusions}
 
 In this work, thermodynamic integration was used to determine the